A detailed explanation of the theory concerning the subpoint method is in the Computing Altitude and True Azimuth category, and in the Celestial Precomputation category. [Figure 11-1] Following is a summary of the steps involved:
- Positively identify the body and measure the altitude using a sextant.
- Because no tabulated information for azimuth or elevation is required for this method, corrections for refraction, parallax, semidiameter, wander error, and sextant correction are applied directly to the Ho.
- The resultant measurement is subtracted from 90° to obtain the co-altitude (co-alt). To convert to NM (1°= 60 NM), multiply the number of degrees times 60. Any fractional portion of degrees is added to the previous value.
Example: Vega is observed at an altitude (Ho) of 88° 23′. Sextant correction is –03′.
88° 23′ – 03′ = 88° 20′
90° – 88° 20′ = 1° 40′
1° 40′ = 60′ + 40′ = 100 NM
In this example, 100 NM represents the distance from the observer’s position to the subpoint of the body. The coordinates of the body are its corresponding declination (Dec) and Greenwich hour angle (GHA). For this example, Vega’s Dec is N38° 46′. The GHA is obtained by applying the sidereal hour angle (SHA) of Vega to the GHA of Aries.
SHA = 080° 59′
GHA Aries = 039° 18′
GHA Vega = 120° 17′
Subpoint of Vega is located at 38° 46′ N l20° 17′ W. The observer is now ready to apply the information:
- Plot the subpoint on an appropriate chart.
- With dividers or compass, span the co-alt distance; in this case 100 NM.
- Use the body’s subpoint (38° 46′ N l20° 17′ W) as the center and 100 NM (co-alt) as the radius. The circle is called the circle of equal altitude and the observer is located on that portion of the circle nearest the dead reckoning (DR) position. There are definite advantages to this method. It requires no precomputation values and plotting is very simple if the observer and body are reasonably close together. When the observer and body are separated by great distances, some disadvantages appear.
- If a body is observed at 20° above the horizon, the observer is 4,200 NM from its subpoint. To swing a LOP from this subpoint, the subpoint and the arc must be plotted on the same chart. To permit plotting of any LOP, the chart must cover an area extending more than 4,000 miles in every direction from the DR position. This means that the chart must be either of such large size that it cannot be spread out on a table in the aircraft, or of such small scale that plotting on it is inaccurate. To cover an area 8,000 miles across, a chart 4 feet square must be drawn to a scale of about 1:10,000,000. Furthermore, measuring would be difficult because of distortion.
- Since a celestial LOP cannot always be drawn by the subpoint method, the intercept method, based on the same principles, is often used.
You can eliminate the need for plotting the body’s subpoint and still draw the arc representing the circle of equal altitude. [Figure 11-2] By using the following formula, you can calculate the altitude and azimuth of the body for the DR position:
Hc = SIN-1 [SIN (DEC’) SIN (LDr) +COS (DEC’)COS (LDr) COS (LHA)]
Z = COS (Z) = [SIN (DEC’) – SIN (LDR) SIN (HC)]/[COS (Hc COS (LDr)]
Zn = Z if SIN (LHA) < 0
Zn = 360 – Z if SIN (LHA) > 0
The calculations may be performed quickly using a programmable calculator, or they may be extracted from the appropriate volume of the National Imagery and Mapping Agency’s Sight Reduction Tables for Air Navigation in a publication referred to as Pub. No. 249. This method enables the observer to use any of the navigational bodies available at the appropriate fix time. Here is a brief review:
- Compute a DR for the time of the position, using preflight or inflight data.
- Determine the necessary entering values for the Pub. 249 volume being used (Lat, LHA, Dec contrary, or same) and extract all the necessary values of computed altitude (Hc) and azimuth angle (Z).
- After making all the necessary conversions and corrections (Chapter 10), compare the Ho and corrected Hc. This difference is the intercept. If the Ho equals the corrected Hc, then the circle of equal altitude passed through the plotting position. If the Ho is greater than the Hc, the difference is plotted in the direction of the true azimuth (Zn). The Zn represents the azimuth from the observer’s position to the subpoint. If the Ho is less than the Hc, plot the difference 180° from the Zn.
- NOTE: If HO is MOre, plot TOward the subpoint (HO MO TO)
Example: The assumed position is 38° N, 121° 30′ W for a shot taken at 1015Z on Aldebaran. The Ho is 32° 14′. The Hc is determined to be 32° 29′ and the Zn is 120°. A comparison of Ho and Hc determines the intercept to be 15 NM away (15A).
Plotting LOP Using Zn Method
- Plot the assumed position and set the intercept distance on the dividers. [Figure 11-3]
- Draw a dashed line through the assumed position toward the subpoint.
- Span intercept distance along dashed Zn line.
- Place plotter perpendicular to Zn.
- Draw LOP along plotter as shown in Figure 11-3.
Plotting LOP Using Flip-Flop Method
- Plot the assumed position and set the intercept distance on the dividers. [Figure 11-4]
- Measure 120° of the Zn with point A of the dividers on the assumed position and place point B of the dividers down. In this case, away from 120° or in the direction of 300° from the assumed position. Slide the plotter along the dividers until the center grommet and the 100/200-mile mark are lined up directly over point B of the dividers marking the intercept point.
- Remove point A of the dividers from the assumed position, keeping point B in place. Flip point A (that was on the assumed position) across the plotter, at the same time expanding the dividers so that point A can be placed on the chart at the 90°/270° mark of the plotter.
- Flop the plotter around and place the straight edge against the perpendicular, which is established by the dividers.
- Draw LOP along the plotter as shown in Figure 11-4.
When using the intercept method, remember:
- For some assumed position near the DR position, find the Hc and Zn of this body for the time of the observation. This is done with the aid of celestial tables, such as Pub. No. 249 or a programmable calculator.
- Obtain needed corrections, sextant correction, refraction, etc., and apply these to the Hc by reversing the sign. Remember, we are striving to derive a precomputed value to ensure the correct body is shot. Measure the altitude (Ho) of the celestial body with the sextant and record the midtime of the observation.
- Find the intercept, which is the difference between Ho and Hc. Intercept is toward the subpoint if Ho is greater than Hc, and away from the subpoint if Ho is smaller than Hc.
- From the assumed position, measure the intercept toward or away from the subpoint (in the direction of Zn or its reciprocal) and locate a point on the LOP. Through this point, draw the LOP perpendicular to the Zn.
Additional Plotting Techniques
The preceding techniques involve the basic plotting procedures used on most stars and the bodies of the solar system. However, there are certain techniques of plotting that are peculiar to their own celestial methods; for example, the plotting of LOPs obtained by using Polaris, which is discussed later. Also, certain precomputation techniques lend themselves more readily to other plotting techniques, such as preplotting the true azimuths or plotting the fix on the DR computer.
These last plotting techniques are discussed in Pub. No. 249 in the section on precomputation. Other special techniques are discussed in the section on curves, in which the celestial observation is plotted on a graph rather than on the chart.